High-Dimensional Convexity, Isoperimetry and Concentration via a Riemannian Vantage Point

Project Information

CONC-VIA-RIEMANN

Grant agreement ID: 637851

DOI

10.3030/637851

Project closed

EC signature date 4 June 2015

Start date 1 October 2015

End date 30 September 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 194 190,00

EU contribution

€ 1 194 190,00

Coordinated by

TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY
Israel

Periodic Reporting for period 4 - CONC-VIA-RIEMANN (High-Dimensional Convexity, Isoperimetry and Concentration via a Riemannian Vantage Point)

Reporting period: 2020-04-01 to 2021-09-30

Summary of the context and overall objectives of the project

The main motif of our project is to emphasize how the vantage point of Riemannian Geometry can be used to enhance our qualitative and quantitative understanding of isoperimetric, functional and concentration properties of convex bodies and other structures in Euclidean space. Understanding these properties is very challenging because high-dimensional space tends to contradict our intuition: for instance, most of the mass of a high-dimensional ball lies almost entirely near the ball's boundary, and not near its center. The situation with more general convex bodies is far less understood, and quantitatively determining where most of their mass lies is important for various applications: creating algorithms to tell convex bodies apart from a ball, understanding the distribution of a point randomly selected inside the body, deciding how to most efficiently cut the convex body in half, etc. Such questions lie at the very heart of the theory of Asymptotic Geometric Analysis and Convex Geometry, and serve as tools in other fields such as Probability Theory, Statistical Physics, Learning Theory and Algorithmic Geometry.

In accordance with our own experience prior to the commencement of the project, it turns out that broadening the scope and incorporating tools from the Riemannian world has indeed led us to significant progress in our understanding of the qualitative and quantitative structure of isoperimetric minimizers in the purely Euclidean setting. Moreover, it turned out that the Riemannian approach has enabled us to advance our understanding in even more general settings, such as in affine differential geometry, sub-Riemannian manifolds and very general metric-measure spaces, as described in the next subsection.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We commenced the project by finding a unified framework for deducing numerous functional inequalities on a weighted Riemannian manifold and its boundary satisfying the Curvature Dimension (CD) condition. These inequalities are known to be intimately connected to the Brunn--Minkowski inequality, a fundamental inequality in Convexity Theory which lies at the heart of our project. Armed with these new tools in the Riemannian setting, we obtained new concentration results in the purely Euclidean setting. Our idea was to change the Euclidean metric to a more general Riemannian one, and perform our analysis on the resulting weighted Riemannian manifold.

We continued to address a strengthened conjectured version of the classical Brunn-Minkowski inequality for origin-symmetric convex bodies. We confirmed that such a (local) strengthening does exist, but with non-optimal magnitude. It turns out that this problem is equivalent to a spectral minimization problem of a certain second-order differential operator introduced by Hilbert on appropriate weighted Riemannian manifold. In addition, we were able to improve the best-known stability estimates for the Brunn--Minkowski and anisotropic isoperimetric inequalities, two important objects of study in our project.

We resolved the isoperimetric conjecture of Kannan, Lovász and Simonovits, a central question in our project, on the subclass of generalized Orlicz balls, thereby significantly enlarging the class of convex bodies for which the conjecture has been confirmed.

It turned out that the tools we developed enabled us to go beyond the Riemannian setting, and we were able to establish sharp Poincaré inequalities on metric-measure (mm) spaces satisfying the MCP condition. We subsequently extended these results to all L^p-Poincaré and log-Sobolev estimates for even more general mm-spaces satisfying the Quasi Curvature Dimension condition, which we introduced, yielding almost optimal results on the Heisenberg group and other sub-Riemannian manifolds.

In the second half of our project, our Riemannian vantage-point enabled us to resolve several long-standing open problems:
- a conjecture of Sturm and Villani regarding whether their CD condition on very general mm-spaces enjoys the local-to-global property.
- the Gaussian double-bubble and more generally multi-bubble isoperimetric conjectures - see the attached figure for the optimal partition of Gauss space on R^2 into three parts and on R^3 into four parts.
- a conjecture of Lutwak on the isoperimetric minimizers of the affinely-invariant quermassintegrals in the higher-rank case.

Towards the end of the project, we continued to study the affine-invariant setting, revealing a novel connection between the Brunn-Minkowski theory and affine differential geometry. We were thus able to resolve the isomorphic version of the log-Brunn-Minkowski conjecture, as well as resolving the conjecture in full for convex bodies with appropriately pinched curvature.

Dissemination:
I have given 32 talks at different venues in Europe, USA and Israel, to an overall estimated audience of well over 1000 people.
I have written 16 project-related papers which are all open-access and available to the public.
I have had 4 post-docs and 1 PhD Student, and we have been conducting periodic meetings and discussions regarding various project-related Mathematics; these young researchers have written an additional 11 project-related papers and 1 Ph.D. Thesis. All of my post-docs are nowadays Assistant Professors in departments around the world.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

All of the above mentioned works constitute original research which go well beyond the previously known state of the art:
- our discovery of a novel flow on Riemannian manifolds which is the analogue of Minkowski addition, a purely linear operation.
- our unified approach for proving Brascamp--Lieb inequalities on manifolds with non-negative (generalized) Ricci curvature works in a variety of more general conditions than was previously considered, such as for negative generalized dimension.
- our idea to obtain Poincaré and log-Sobolev estimates on log-concave measures on convex domains in Euclidean space by introducing various types of Riemannian metrics, and in particular conformal metrics, on these domains.
- our confirmation of the KLS conjecture on generalized Orlicz balls.
- our resolution of the long-standing Sturm-Villani local-to-global conjecture for the Curvature-Dimension condition on metric-measure spaces.
- our interpretation of the L^p-Brunn-Minkowski conjecture as a next-eigenvalue estimate for a second-order differential operator introduced by Hilbert allowed us to (locally) resolve the conjecture for a range of p's going strictly below the classical p=1 for the very first time.
- our resolution of the Gaussian double-bubble and multi-bubble conjectures, which was the first time that an optimal isoperimetric partition of any space of dimension greater than two into four or more parts was established.
- our resolution of Lutwak's isoperimetric conjecture for the affine quermassintegrals.
- in our work on the log-Minkowski problem, we revealed a connection of the Brunn-Minkowski theory to centro-affine differential geometry which has escaped convex geometers for over a century.

Conjectured Gaussian double-bubble in R^2 and triple-bubble in R^3